"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Archive for the ‘math.RA’ Category

Let be a commutative ring and let be a -algebra. In this post we’ll investigate a condition on which generalizes the condition that is a finite separable field extension (in the case that is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over .”

Definition-Theorem: The following conditions on are all equivalent, and all define what it means for to be a separable-algebra:

is projective as an -bimodule (equivalently, as a left -module).

The multiplication map has a section as an -bimodule map.

admits a separability idempotent: an element such that and for all (which implies that ).

(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless is a field or is projective over . It’s been removed since it plays no role in the post anyway.)

When is a field, this condition turns out to be a natural strengthening of the condition that is semisimple. In general, loosely speaking, a separable -algebra is like a “bundle of semisimple algebras” over .

Previously we suggested that if we think of commutative algebras as secretly being functions on some sort of spaces, we should correspondingly think of cocommutative coalgebras as secretly being distributions on some sort of spaces. In this post we’ll describe what these spaces are in the language of algebraic geometry.

Let be a cocommutative coalgebra over a commutative ring . If we want to make sense of as defining an algebro-geometric object, it needs to have a functor of points on commutative -algebras. Here it is:

.

In words, the functor of points of a cocommutative coalgebra sends a commutative -algebra to the set of setlike elements of . In the rest of this post we’ll work through some examples.

Mathematicians are very fond of thinking about algebras. In particular, it’s common to think of commutative algebras as consisting of functions of some sort on spaces of some sort.

Less commonly, mathematicians sometimes think about coalgebras. In general it seems that mathematicians find these harder to think about, although it’s sometimes unavoidable, e.g. when discussing Hopf algebras. The goal of this post is to describe how to begin thinking about cocommutative coalgebras as consisting of distributions of some sort on spaces of some sort.

Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”

Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor , which a priori takes values in algebras (it’s left adjoint to the forgetful functor from algebras to Lie algebras), in fact takes values in Hopf algebras. This upgraded functor continues to be a left adjoint, although the forgetful functor is less obvious. Given a Hopf algebra , its primitive elements are those elements satisfying

where is the comultiplication. The primitive elements of a Hopf algebra form a Lie algebra, and this gives a forgetful functor from Hopf algebras to Lie algebras whose left adjoint is the upgraded universal enveloping algebra functor.

The key observation is that this upgraded functor is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms and Hopf algebra homomorphisms . This is more or less equivalent to the claim that the natural inclusion induces an isomorphism from to the Lie algebra of primitive elements of , which can be proven using the PBW theorem.

Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is not true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)

Let be a ring. Previously we characterized the finitely presented projective (right) -modules as the tiny objects in : the objects such that

preserves colimits. We also highlighted the key role that these modules play in Morita theory.

If is a commutative ring, then has a natural symmetric monoidal structure which allows us to describe another finiteness condition called dualizability. Unlike tininess, dualizability makes no reference to colimits; instead, it is a purely equational definition involving the monoidal structure. The dualizable modules are again the finitely presented projective -modules.

Dualizability implies that we can treat finitely presented projective -modules like finite-dimensional vector spaces in many ways: for example, dualizability allows us to define the trace of an endomorphism. Moreover, since dualizability is defined using only a monoidal structure, it makes sense in very general settings, and we’ll look at some more exotic examples of dualizable objects as well.

Duals are also a special case of a 2-categorical notion of adjunction which, in the 2-category of categories, functors, and natural transformations, reproduces the usual notion of adjunction. In a suitable 2-category it will also reproduce another characterization of finitely presented projective modules, this time over noncommutative rings.

Previously we proved a theorem due to Gabriel characterizing categories of modules as cocomplete abelian categories with a compactprojectivegenerator, where “generator” meant “every object is a colimit of finite direct sums of copies of the object.”

But we also used “generator” to mean “every object is a colimit of copies of the object,” and noted that these conditions are not equivalent: as this MO question discusses, the abelian group satisfies the first condition but not the second. More generally, as Mike Shulman explains here, there are in fact many inequivalent definitions of “generator” in category theory.

The goal of this post is to sort through a few of these definitions, which turn out to be totally ordered in strength, and find additional hypotheses under which they agree. As an application we’ll restate Gabriel’s theorem using weaker definitions of “generator” and give a more explicit description of all of the rings Morita equivalent to a given ring.

Let be an abelian group and be a collection of endomorphisms of . The commutant of is the set of all endomorphisms of commuting with every element of ; symbolically,

.

The commutant of is equal to the commutant of the subring of generated by the , so we may assume without loss of generality that is already such a subring. In that case, is just the ring of endomorphisms of as a left -module. The use of the term commutant instead can be thought of as emphasizing the role of and de-emphasizing the role of .

The assignment is a contravariant Galois connection on the lattice of subsets of , so the double commutant may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.